1.2.3 Relationships between variables

Homeownership rate and multi-unit structures

• homeownership: the percentage of homes that are owned by residents
• multi-unit: the percentage of housing units that are in multi-unit structures (e.g., apartments, condos)

Are these two variables related?

Associated Variables

• The multi-unit and homeownership rates are said to be associated because the plot shows a discernible pattern.
  • The downward trend means the variables are negatively associated.
• When two variables show some connection with one another, they are called associated variables.
• If two variables are not associated, then they are said to be independent. That is, two variables are independent if there is no evident relationship between the two.

1.2.4 Explanatory and response variables

Suppose that \(X\) and \(Y\) are associated variables.

• If variable \(X\) helps us explain or predict the value of variable \(Y\), we say that \(X\) is the explanatory variable and \(Y\) is the response variable.
• Sometimes (not always) the explanatory variable affects the response variable, i.e., the change in one variable causes a change in the other.
  • Example: Does the median household income in a county cause its population size to change?

1.2.5 Observational studies and experiments

There are two primary types of data collection: experiments and observational studies.

• In an experiment, researchers put subjects in two or more groups and compare them.
  • In a randomized experiment, researchers randomly assign the groups. (e.g., stent study)
• In an observational study, researchers collect data in a way that does not directly interfere with how the data arise. (e.g, county data set).
• Beware: Association \(\neq\) Causation.
  • Observational studies cannot determine causation (e.g., TV’s predict life expectancy)
  • Well-designed randomized experiments can prove causation.

Air pollution

Researchers collected data to examine the relationship between air pollutants and preterm births in Southern California. During the study air pollution levels were measured by air quality monitoring stations. Specifically, levels of carbon monoxide were recorded in parts per million, nitrogen dioxide and ozone in parts per hundred million, and coarse particulate matter (PM\(_{10}\)) in \(\mu g/m^3\). Length of gestation data were collected on 143,196 births between the years 1989 and 1993, and air pollution exposure during gestation was calculated for each birth. The analysis suggested that increased ambient PM\(_{10}\) and, to a lesser degree, CO concentrations may be associated with the occurrence of preterm births.

Migraines and acupuncture

A migraine is a particularly painful type of headache, which patients sometimes wish to treat with acupuncture. To determine whether acupuncture relieves migraine pain, researchers conducted a randomized controlled study where 89 individuals who identified as female diagnosed with migraine headaches were randomly assigned to one of two groups: treatment or control. Forty-three (43) patients in the treatment group received acupuncture that is specifically designed to treat migraines. Forty-six (46) patients in the control group received placebo acupuncture (needle insertion at non-acupoint locations). Twenty-four (24) hours after patients received acupuncture, they were asked if they were pain free. Results are summarized in the contingency table below. Also provided is a figure from the original paper displaying the appropriate area (M) versus the inappropriate area (S) used in the treatment of migraine attacks.

1. What percent of patients in the treatment group were pain free 24 hours after receiving acupuncture?

2. What percent were pain free in the control group?

3. In which group did a higher percent of patients become pain free 24 hours after receiving acupuncture?

4. What are the explanatory and response variables in this study? Classify each as numerical or categorical.

5. Your findings so far might suggest that acupuncture is an effective treatment for migraines for all people who suffer from migraines. However, this is not the only possible conclusion. What is one other possible explanation for the observed difference between the percentages of patients that are pain free 24 hours after receiving acupuncture in the two groups?